Theres been this fake (steal) it till you make it, wild west approach to growth. Youtube, Buzzfeed, Imgur. You just host anybodys content regardless of if the poster is the owner, and once you get to scale, then you handle copyright and creating your own content so you arent as dependent on external creators.
But in Giphys case, they never have to take the extra step. Because they are so short, they are much more likely to pass fair use, and they can just host anybodys anything, barring some illegal fringes, without having to pay for the rights.